No. Your opponent cannot cast any spells, activate any abilities, or do any other actions while you have priority, unless specifically directed to do so.
116.1. Unless a spell or ability is instructing a player to take an action, which player can take actions at any given time is determined by a system of priority. The player with priority may cast spells, activate abilities, and take special actions.
116.3a The active player receives priority at the beginning of most steps and phases, after any turn-based actions (such as drawing a card during the draw step; see rule 703) have been dealt with and abilities that trigger at the beginning of that phase or step have been put on the stack. No player receives priority during the untap step. Players usually don’t get priority during the cleanup step (see rule 514.3)
116.3d If a player has priority and chooses not to take any actions, that player passes. If any mana is in that player’s mana pool, he or she announces what mana is there. Then the next player in turn order receives priority.
Since the Goblin Grenade has "sacrifice a goblin" as an additional cost to casting the spell, the goblin will not be on the battlefield, which means it will be an illegal target for Unsummon
117.8. Some spells and abilities have additional costs. An additional cost is a cost listed in a spell’s rules text, or applied to a spell or ability from another effect, that its controller must pay at the same time that player pays the spell’s mana cost or the ability’s activation cost. A cost is an additional cost only if it’s phrased using the word “additional.” Note that some additional costs are listed in keywords; see rule 702.